Nonlinear Kinematic Relations Research Articles

Nonlinear large-amplitude vibration analysis of annular sector plates made of FGMs subjected to cooling shock

A numerical approach is developed in the present article to study the geometrically nonlinear vibrations of annular sector plates made of functionally graded materials (FGMs) due to being exposed to cooling shock. The first-order shear deformation plate theory (FSDT) is used in the modeling of plate, and the governing equations of motion are derived based on Hamilton's principle considering von Kármán nonlinear kinematic relations. It is also considered that the properties of FGMs are dependent on temperature and distribution of materials. According to the uncoupled thermoelasticity theory, the temperature distribution is obtained via 1D Fourier-type transient heat conduction equation. To numerically solve the problem, the generalized differential quadrature (GDQ) method and Newmark-beta integration scheme are utilized. Considering two different types of thermal loading as cooling shock, the effects of various parameters including thermal load rapidity time, geometry, magnitude of thermal load and material properties on the large-amplitude vibrations of annular sector plates are investigated.

Closed form solutions for thermo-mechanical buckling analysis of shallow piezo-laminated spherical shells

Thermo-mechanical buckling analysis of laminated spherical shells bounded with the piezoelectric actuator layers has been presented in this article. Applied loads are a combination of thermal, mechanical and electrical loads. Derivations of equilibrium and stability equations are based on the classical laminated shell theory and Sanders’s nonlinear kinematic relations. The analysis employs the Galerkin method to achieve closed-form formulas for the buckling loads of the piezo-laminated spherical shell. Effects of applied actuator voltage, voltage sign, geometrical parameters of the shell (length, diameter, and thickness), material properties, and layout of composite layers on the buckling strength are investigated. Analytical solutions are obtained for three types of thermal loading, and one thermo-mechanical loading condition. The results are validated by the known data in the literature.

Stress concentration effect on deflection and stress fields of a master leaf spring through domain decomposition and geometry updation technique

Abstract Theoretical and experimental large deflection and stress analysis of a master leaf spring considering stress concentration effect of clamping is reported. The non-uniformly curved master leaf spring under three point bending subjected to moving boundaries is modeled. Geometrically nonlinear strain-displacement relations, as necessary for the theoretical analysis, are derived through visualization of physics behind the large deformation problem. An embedded curvilinear coordinate system is considered, to study the combined effects of non-uniform curvature, bending, stretching and shear deformation including cross-sectional warping. Governing equation of the non-uniformly curved beam system is derived in variational form using energy method, based on linear material constitutive relations and the derived nonlinear kinematic relations. An iterative solution scheme through successive geometry updation is developed and executed in MATLAB® software to solve the governing equation involving strong geometric nonlinearity together with complicating moving boundary effect. Experimental deflection profiles under static loading are obtained through manual image processing technique using AutoCAD® software. Whereas, strain measurements are performed using strain gauges with data acquisition system (HBM-MX840B). Comparison between the theoretical and experimental results lead towards observation on stress concentration effect due to presence of geometric discontinuity in form of a small hole in the physical system. A modified formulation is proposed using domain decomposition method incorporating effect of geometric discontinuity through an equivalent curved beam geometry of variable cross-section. The modified theoretical model is validated successfully with the experimental results, and observations on stress characteristics and effect of hole diameter to beam width ratio are made.

Nonlinear mechanics of fragmented beams

This paper presents an analytical model for the nonlinear analysis of fragmented beams — beams which are assembled from separate fragments (blocks) without any binding material or connector. The structural integrity in these structures is provided by the shapes of the fragments and the axial compression applied at the boundaries. However, this compression also introduces a destabilizing mechanism in the form of P-Delta effect, which may lead to a loss of the global stability. In addition, the absence of binding allows the fragments to move and rotate independently, which causes detachments between them. In this paper, this coupled nonlinear behavior is thoroughly studied by applying geometrically nonlinear kinematic relations at the fragments and nonlinear traction laws at the interfaces between them. In order to capture the localization of stresses in the partially separated fragments, the high-order beam theory is implemented. The obtained nonlinear governing equations are solved numerically and the results are compared with ones obtained by finite element simulations. It is demonstrated that the progressive detachment between the fragments yields a localization of stresses at the contact areas. These detachments and stress localizations impair the bending stiffness of the beam, and together with the axial compression, present a destructive coupled mechanism. This mechanism controls the strength and stability of the fragmented beam. Finally, a parametric study on the effects of the axial compression and boundary conditions on the strength and energy absorption capacity of the beams is presented.

Fractional-order structural stability: Formulation and application to the critical load of nonlocal slender structures

This study presents a framework to perform stability analysis of nonlocal solids whose behavior is described according to the fractional-order continuum theory. In this formulation, space fractional-order operators are used to capture the nonlocal response of the medium by means of nonlocal kinematic relations. We use the geometrically nonlinear fractional-order kinematic relations within an energy based approach to establish the Lagrange-Dirichlet stability criteria for nonlocal structures. This energy based approach to nonlocal structural stability is possible due to a positive-definite and thermodynamically consistent definition of the deformation energy enabled by the fractional-order kinematic formulation. The Rayleigh-Ritz coefficient for critical load is also derived for linear buckling conditions. The fractional-order formulation is finally used to determine critical loads for buckling of the slender nonlocal beams and plates using a dedicated fractional-order finite element solver. Results establish that, in contrast to existing studies, the effect of nonlocal interactions is observed on both the material and the geometric stiffness, when using the fractional-order kinematics approach. These observations are supported quantitatively via the solution of case studies that focus on the critical buckling response of fractional-order nonlocal slender structures, and a direct comparison of the fractional-order approach with classical nonlocal approaches.

Nonlinear finite element study on forced vibration of cylindrical micro-panels based on modified strain gradient theory

This research deals with the nonlinear forced vibration analysis of cylindrical micro-panels (CMP) under the modified strain gradient shell theory using the higher-order finite element method. The main objective is to study the impacts of length scale and geometrical parameters on the forced vibration behavior of CMP. The mechanical model based on the modified strain gradient theory (SGT), shear deformation shell theory (SDST), and von Kármán nonlinear kinematic relations is developed using Hamilton’s principle and the hermitian finite elements are employed to solve the problem. Variations of the frequency responses are presented to analyze the nonlinear forced vibration of CMP.

Nonlinear strain gradient forced vibration analysis of shear deformable microplates via hermitian finite elements

Based on the modified strain gradient model (MSGM) and Mindlin-Reissner (M-R) plate theory, the geometrically nonlinear forced vibration of rectangular microplates via finite element (FE) method is investigated in this research. The nonlinear mathematical model of the problem is provided under the von Karman nonlinear kinematic relations within the M-R plate theory along with the MSGM. By the use of Hamilton’s principle and fully confirming rectangular hermitian finite elements, the stiffness and mass matrices, as well as the force vector, are obtained. Various numerical examples are represented to verify the efficiency and validity of the proposed model. By reducing the constitutive relations, the results for the modified couple stress model (MCSM) are also presented. Various numerical examples are given to investigate the impacts of length-scale parameter and geometrical parameters on the nonlinear forced vibration responses.

Nonlinear lay-up optimization of variable stiffness composite skew and taper cylindrical panels in free vibration

In the present study, the fundamental natural frequencies of curvilinear fiber composite skew and taper cylindrical panels are optimized applying genetic algorithm (GA). Later, the fundamental amplitude-dependent nonlinear frequency behavior of the optimized curved fiber layup configurations is studied and compared with the reference unidirectional fiber layup. The variable stiffness behavior is obtained by altering the fiber angles continuously according to the curvilinear fiber path function in the composite laminates. A nonlinear structural model is utilized based on the virtual work principle. Green’s nonlinear kinematic strain relations are used to account for the geometric nonlinearities and the first-order shear deformation theory (FSDT) is adopted to generalize the formulation for the case of moderately thick cylindrical panels including transverse shear deformations. The goal is to determine how the variable stiffness parameters affect the linear and nonlinear free vibration behavior of the skew and taper cylindrical panels. Consequently, one may find the optimum fiber path with improved structural characteristics for the cylindrical panel. Eight-layered composite skew and taper cylindrical panels at two different boundary condition sets are considered in this research. Generalized Differential Quadrature (GDQ) method of solution is employed to solve the nonlinear governing equations of motion. Numerical results demonstrate the degree of effectiveness for fiber angle paths, boundary conditions, and geometrical non-uniformities on the fundamental frequencies of the cylindrical panel. Eventually, optimum fiber angles of each layer in free vibration analysis are presented.

Nonlinear vibration analysis of functionally graded GPL-RC conical panels resting on elastic medium

The detailed investigation on the nonlinear vibration analysis of functionally graded graphene platelets-reinforced composites (FG GPL-RC) conical panels on the elastic medium is presented. The nonlinear governing equations of nanocomposite panels are derived following the first-order shear deformation shell theory (FSDST) and the von Kármán nonlinear kinematic relations using Hamilton's principle, while the overall mechanical properties of the composite panel are estimated through the refined Halpin-Tsai approach. The numerical solution based on the 2-D differential quadrature method (DQM), arc-length continuation technique and the harmonic balance approach is presented to solve the problem numerically and find the frequency response. A diverse range of results is delivered to show the effect of essential parameters such as volume fraction and distribution pattern of GPLs, semi-vertex angle, elastic foundation and boundary supports on the nonlinear vibrational of nanocomposite conical panels.

On the specifics of behavior of the sandwich plate composite facing layers under local loading

The problem of a four-point bending of sandwich plates with external layers of a fiber reinforced plastic is considered, the results of numerical and experimental studies are presented. It is shown that in the vicinity of the loading roller, which exerts a local effect on the external layer, there is a strong decrease in the transverse shear secant modulus of fiber reinforced plastic with an increase in the transverse shear strains. The numerical solution of the problem of the plate bending is carried out in a physically and geometrically nonlinear formulation using various relations of the finite element method, two variants of geometrically nonlinear kinematic relations of the equations of the elasticity theory and different variants of the loading process parameter. Along with the classical nonlinear relations, the problem solutions are also constructed on the basis of consistent relations between strains and displacements, the use of which allows one to avoid the appearance occurrence of false bifurcation points. The results of the numerical calculations obtained with different methods and use of different ratios are presented, the analysis of which showed their small difference. It was revealed that the stability loss and failure of the external layers of a sandwich plate occurs due to the stability loss in the nonclassical transverse-shear mode. To determine the ultimate load, which is accompanied with a loss of strength of the external loaded layer, the Tsai-Wu criterion was used. A comparative analysis of the behavior of the plate external layers at different thicknesses and different diameters of the loading roller is carried out. It is shown that the ultimate load is practically not affected by the roller diameter, while the load at which the external layer loses its stability, is very sensitive to a change in its value.

Modeling of interfacial crack propagation and kinking in sandwich panels

The paper presents an analytical model for the nucleation, propagation, and kinking of cracks in sandwich panels. The analytical approach adopts the variational principle of minimum of the total potential energy and it is based on cohesive interface modeling. The mathematical modeling considers two face sheets, a core layer, and an interfacial core-face sheet crack that can kink into the core and form a diagonal shear crack. This phenomenon, which has been observed in experiments, divides the core into two bodies. The core is therefore modeled using such two body approach. Three interfaces are introduced between the four components and appropriate interfacial laws are used to describe the interfacial phenomenon. The First-Order Shear Deformation Theory with geometrically nonlinear kinematic relations is adopted for the face sheets. The two core layers use linear geometrical relations and assume the high order polynomial displacement fields of the Extended High-Order Sandwich Panel Theory. Linear elastic laws are considered for the two face sheets and the core. The model is validated through a comparison with experimental observations of pre-cracked Double Cantilever Beam sandwich specimens that show a diversity of crack paths. The comparison and the quantitative investigation explore the capabilities and the limitations of the approach and shed light on the nucleation, debonding, and kinking phenomena in soft-core sandwich panels.

Effects of geometric nonlinearity on the pull-in instability of circular microplates based on modified strain gradient theory

Presented in this study is an analytical investigation on the size-dependent nonlinear vibration and pull-in instability of circular microplates subjected to the electrostatic, Casimir, and hydrostatic forces. Based on the modified strain gradient theory in conjunction with the Kirchhoff thin plate theory and von Kármán’s nonlinear kinematic relations, the governing equations were derived using the variational principle. The Galerkin technique (GT) and Homotopy analysis method (HAM) are employed to present the analytical solution considering the clamped boundary condition. Different comparative studies are presented to show the accuracy of the model. As the main novelty of this study, the effects of the geometric nonlinearity on the strain gradient dynamic pull-in instability of circular microplates are presented through a wide range of analytical results. It is observed that by increasing the gap distance, the impacts of nonlinear strains on pull-in behavior become more remarkable.

Geometrical Nonlinear Effects in the Dynamic Response of Sandwich Beams

AbstractThe geometrical nonlinear dynamic response of sandwich beams is studied using a dynamic high-order nonlinear model. The model is derived using the variational principle of virtual work and uses the Extended High-Order Sandwich Panel Theory approach with consideration of two interfaces between the three layers. A first-order shear deformation theory is adopted for the face sheets, while the kinematic assumption of high-order small deformations that account for out-of-plane compressibility are considered for the core layer. The nonlinearity of the dynamic model is introduced by considering geometrically nonlinear kinematic relations in the face sheets. The nonlinear kinematic relations and the dynamic modeling aim to evaluate the effects of the two features and their coupling on the response. The nonlinear dynamic response of sandwich beams is studied through two numerical cases and comparison of the nonlinear results with their linear counterparts. The first case looks into the coupling of the global geometrical nonlinear behavior with the dynamic behavior. The second case focuses on the local instability of the face sheets and its interaction with the compressibility of the core in the dynamic response of soft core sandwich beams. The comparison of linear and nonlinear dynamic response in the two cases sheds light on the coupling of the geometrical nonlinear and dynamic behaviors. Among other features, the latter is expressed by nonlinear attractors, higher modes response, nonlinear frequency response, and significant wrinkling response.

Influences of surface effects on large deflections of nanomembranes with arbitrary shapes by the coupled BE-RBFs method

This paper aims to analyze large deflections of nanomembranes including surface effects with arbitrary shapes. The nonlinear differential equations of nanomembranes are formulated by using the nonlinear kinematic relations and the surface elasticity theory of Gurtin–Murdoch. The principle of virtual work is applied to establish the three governing partial differential equations of nanomembranes. The coupled boundary element-radial basis functions (BE-RBFs) method is developed to solve the complicated nonlinear problem of nanomembranes. The proposed methodology is based on the analog equation method in conjunction with radial basis functions in order that the boundary line integrals and boundary elements are only involved. The validation and accuracy of the present method are evaluated by comparing the obtained results with those available from other numerical solutions. The proposed formulation can provide the numerical results that correspond to the experimental findings of the monolayer circular graphene membrane by specifying the proper surface properties. The influences of the surface elastic constants and residual surface stress on large deflection responses of nanomembranes are evidently investigated. Moreover, some numerical results of the present formulations could serve as a benchmark for the numerical evaluation of future research. Finally, the interesting results of large deflection analysis of the various nanomembrane shapes using the coupled BE-RBFs method are highlighted.

Post-buckling analysis of non-uniformly heated functionally graded cylindrical shells enhanced by shape memory alloys using classical lamination theory

Thermal post-buckling analysis of functionally graded cylindrical shells enhanced by shape memory alloys under uniform and non-uniform heating is presented in this article. Nonlinear equilibrium equations are derived based on the classical lamination theory and von-Karman nonlinear kinematic relations and post-buckling field is investigated using Galerkin method. For temperature dependency of material properties, a numerical solution is applied to solve the nonlinear equilibrium equation using finite difference method to solve the nonlinear heat conduction equation and layered model to evaluate the thermal stress of hybrid cylindrical shells. A closed-form solution is also presented for temperature independency of material properties. Brinson model is adopted to describe the thermo-mechanical behavior of shape memory alloys. Numerical results are presented for evaluating the effects of shape memory alloy layer and functionally graded material cylindrical shells properties on suppressing of the post-buckling path of hybrid cylindrical shells.

Hygro-Thermal Nonlinear Analysis of a Functionally Graded Beam

Nonlinear behavior of a functionally graded cantilever beam is analyzed under non-uniform hygro-thermal effect. To solve this problem, finite element method is applied within plane solid continua. Total Lagrangian approach is utilized in the nonlinear kinematic relations. Newton-Raphson method with incremental displacement is used in nonlinear solution. Comparison study is performed. Effects of material distribution, temperature and moisture changes on nonlinear deflections of the functionally graded beam are presented and discussed.

Post-buckling analysis of a fiber reinforced composite beam with crack

In this paper, the post-buckling analysis of a cracked fiber reinforced composite (FRC) beam is investigated under a compressive load. In the solution of the problem, the finite element method is used within the first shear beam theory. The total Lagrangian approach is used in the nonlinear kinematic relations. The crack is modelled as a rotational spring. In the nonlinear solution of the post-buckling problem, the Newton-Raphson method is used. The effects of the fibre orientation angles, the volume fraction of the fibre, the crack depth and the crack locations on the post-bucking responses of the FRC beams are investigated in numerical results. The results show that the crack depth, the fibre orientation angles and the volume fraction of fibre have important role in the post-buckling responses of the FRC beams.

Hygrothermal Post-Buckling Analysis of Laminated Composite Beams

The main goal in this paper is to make analysis of post-buckling of laminated composite beams under hygrothermal effect. In solution of problem, finite element method is utilized with the first shear beam theory. Total Lagrangian approach is used nonlinear kinematic relations. It is known that post-buckling problems are geometrically nonlinear problems. In nonlinear solution of problem, the Newton–Raphson method is used based on incremental displacement. The novelty in this study is to investigate the hygrothermal post-buckling analysis of laminated composite beams by using total Lagrangian nonlinear approach. The influences of temperature, moisture, fiber orientation angles, stacking sequence of laminas on post-buckling responses of composite laminated beam are illustrated and examined in numerical results. The results show that fiber orientation angles, stacking sequence of laminas play an important role in hygrothermal post-buckling responses of laminated beams. Also, comparison studies are performed with special results of published paper.

Nonlinear behavior of fiber reinforced cracked composite beams

This paper presents geometrically nonlinear behavior of cracked fiber reinforced composite beams by using finite element method with and the first shear beam theory. Total Lagrangian approach is used in the nonlinear kinematic relations. The crack model is considered as the rotational spring which separate into two parts of beams. In the nonlinear solution, the Newton-Raphson is used with incremental displacement. The effects of fibre orientation angles, the volume fraction, the crack depth and locations of the cracks on the geometrically nonlinear deflections of fiber reinforced composite are examined and discussed in numerical results. Also, the difference between geometrically linear and nonlinear solutions for the cracked fiber reinforced composite beams.

Correlation between Geometrically Nonlinear Elastoviscoplastic Constitutive Relations Formulated in Terms of the Actual and Unloaded Configurations for Crystallites

A correct description of severe plastic deformation requires the use of nonlinear kinematic and constitutive relations. The relations should meet certain criteria, such as the frame-independence, closed stress cycles and the absence of energy dissipation under elastic cyclic loading. A most complicated issue in formulating these relations is the decomposition of motion into quasi-rigid and strain-induced motions, which is an extremely difficult problem with no unambiguous solution. For the majority of structural metals and alloys, this decomposition can be done in a physically justified way on the level of crystallites. Earlier, using a multilevel approach we introduced a corotational coordinate system for crystallites responsible for quasi-rigid motion. This allowed us to formulate frame-independent mesoscopic constitutive relations in the actual configuration and to perform test calculations to show that the above criteria are satisfied with high accuracy. A new way of motion decomposition was proposed which is a multiplicative representation of the deformation gradient with an explicit extraction of the corotational frame motion. Elastoviscoplastic constitutive relations satisfying the above criteria were formulated in terms of an unloaded "lattice" configuration. However, it is more preferable to formulate nonlinear boundary value problems in terms of the actual configuration in rates. This paper is aimed to study correlation between the earlier derived constitutive relations formulated in terms of the actual configuration and in terms of the unloaded "lattice" configuration. Example problems are solved to demonstrate the closeness of results obtained with these constitutive relations.